Finite Group Databasebeta

← #189812

Subgroups — #189812

Order 512.

1500 conjugacy classes of subgroups.

The subgroup lattice has 533 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2285C211yesmaximal, characteristic
2#20714C211yesmaximal, characteristic
3C4×#380C211yesmaximal, characteristic
4#8716C214yesmaximal, complemented
5#8712C214yesmaximal
6#8014C214yesmaximal, complemented
7C22×#117—21no—
8C23×C42C2211yescharacteristic
9C2×#431C2212yes—
10#2285—21no—
11#2285—21no—
12#2285—21no—
13#2285—21no—
14#2285C412yescomplemented
15#2285C412yescomplemented
16#2285C2212yes—
17#2285C2212yes—
18C2×#384C411yescharacteristic, complemented
19C2×#384C411yescharacteristic, complemented
20C2×#382—21no—
21C2×#380—21no—
22C2×#380—21no—
23C2×#380C2212yes—
24C2×C4×C4⋊C4—21no—
25C2×C4×C4⋊C4C412yescomplemented
26C2×C4×C4⋊C4C2212yes—
27C2×C4×C22⋊C4—21no—
28#1817C2212yes—
29#1797—21no—
30#1797—21no—
31#1779C2212yes—
32C4×C22⋊C8—21no—
33C4×C22⋊C8—21no—
34C4⋊#95—21no—
35#1434C2212yes—
36C22⋊#95C2212yes—
37C4×#95—21no—
38#1294C22116yes—
39C24×C4D411yescharacteristic
40C24×C4D411yescharacteristic
41C24×C4C2311yescharacteristic
42C2×#117—22no—
43C2×#117—22no—
44C2×#117—22no—
45C2×#117—22no—
46C2×#117—44no—
47C2×#117—44no—
48C22×C4⋊C4—21no—
49C22×C4⋊C4—21no—
50C22×C4⋊C4—21no—
51C22×C4⋊C4—21no—
52C22×C4⋊C4—22no—
53C22×C4⋊C4C2×C412yes—
54C22×C4⋊C4C2×C412yes—
55C22×C22⋊C4—22no—
56C22×C22⋊C4—22no—
57C22×C42—28no—
58C22×C42—22no—
59C22×C42—22no—
60C22×C42C2×C411yescharacteristic
61C22×C42C2×C411yescharacteristic
62C22×C42C2×C411yescharacteristic
63C22×C42C2×C411yescharacteristic
64C22×C42D411yescharacteristic
65C22×C42D411yescharacteristic
66C22×C42D411yescharacteristic
67C22×C42D411yescharacteristic
68C22×C42D411yescharacteristic
69C22×C42D411yescharacteristic
70C22×C42C2311yescharacteristic
71C22×C42C2311yescharacteristic
72#431—28no—
73C2×C4⋊C8—22no—
74C2×C22⋊C8—22no—
75C2×C22⋊C8—22no—
76C2×C22⋊C8C2314yes—
77C2×C4×C8—22no—
78#384—21no—
79#384—21no—
80#384—21no—
81#384—21no—
82#384C2×C412yes—
83#384C2×C412yes—
84#384C2×C412yes—
85#384C2×C412yes—
86#382—22no—
87#382—22no—
88#382—22no—
89#382—22no—
90#380—28no—
91#380—22no—
92#380—22no—
93#380—22no—
94#380—22no—
95#380—22no—
96#380—22no—
97#380C2×C414yes—
98#380C2314yes—
99C4×C4⋊C4—22no—
100C4×C4⋊C4—22no—
101C4×C4⋊C4—22no—
102C4×C4⋊C4—22no—
103C4×C4⋊C4—22no—
104C4×C4⋊C4—22no—
105C4×C4⋊C4—28no—
106C4×C4⋊C4C2×C414yes—
107C4×C4⋊C4C2×C414yes—
108C4×C22⋊C4—44no—
109C2×#95—22no—
110C2×#95—22no—
111C2×#95—22no—
112C2×#95—22no—
113C2×#95—22no—
114C2×#95C2314yes—
115#336—22no—
116#336—22no—
117C25C2×D411yescharacteristic
118C23×C4—24no—
119C23×C4—24no—
120C23×C4—24no—
121C23×C4—24no—
122C23×C4—24no—
123C23×C4—22no—
124C23×C4—22no—
125C23×C4—24no—
126C23×C4—42no—
127C23×C4—42no—
128C23×C4—42no—
129C23×C4C22⋊C411yescharacteristic
130C23×C4C22⋊C411yescharacteristic
131C23×C4C22⋊C411yescharacteristic
132C23×C4C22⋊C411yescharacteristic
133C23×C4C22⋊C411yescharacteristic
134C23×C4C22⋊C411yescharacteristic
135C23×C4C22⋊C411yescharacteristic
136C23×C4C22⋊C411yescharacteristic
137C23×C4D814yes—
138C23×C4SD1614yes—
139C23×C4C22×C411yescharacteristic
140C23×C4C22×C411yescharacteristic
141C23×C4C2×D411yescharacteristic
142C23×C4C2×D411yescharacteristic
143C23×C4C2×D411yescharacteristic
144C23×C4C2×D411yescharacteristic
145C23×C4C2×D411yescharacteristic
146C23×C4C2×D411yescharacteristic
147C23×C4C2×D411yescharacteristic
148C23×C4C2411yescharacteristic
149C22×C8—24no—
150#117—48no—
151#117—48no—
152#117—44no—
153#117—44no—
154#117—44no—
155#117—44no—
156C2×C4⋊C4—28no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—44no—
168C2×C4⋊C4—44no—
169C2×C4⋊C4—44no—
170C2×C4⋊C4—44no—
171C2×C4⋊C4—44no—
172C2×C4⋊C4—44no—
173C2×C4⋊C4—44no—
174C2×C4⋊C4—44no—
175C2×C4⋊C4C22×C414yes—
176C2×C4⋊C4C22×C414yes—
177C2×C22⋊C4—28no—
178C2×C22⋊C4—24no—
179C2×C22⋊C4—24no—
180C2×C22⋊C4—44no—
181C2×C22⋊C4—44no—
182C2×C22⋊C4—44no—
183C2×C22⋊C4—44no—
184C2×C42—24no—
185C2×C42—24no—
186C2×C42—24no—
187C2×C42—24no—
188C2×C42—24no—
189C2×C42—24no—
190C2×C42—24no—
191C2×C42—24no—
192C2×C42—24no—
193C2×C42—24no—
194C2×C42—24no—
195C2×C42—24no—
196C2×C42—24no—
197C2×C42—48no—
198C2×C42—44no—
199C2×C42—44no—
200C2×C42—44no—
201C2×C42C22⋊C411yescharacteristic
202C2×C42C22⋊C411yescharacteristic
203C2×C42C22⋊C411yescharacteristic
204C2×C42C22⋊C411yescharacteristic
205C2×C42C22⋊C411yescharacteristic
206C2×C42C22⋊C411yescharacteristic
207C2×C42C22⋊C411yescharacteristic
208C2×C42C22⋊C411yescharacteristic
209C2×C42C22×C411yescharacteristic
210C2×C42C22×C411yescharacteristic
211C2×C42C22×C411yescharacteristic
212C2×C42C22×C411yescharacteristic
213C2×C42C2×D411yescharacteristic
214C2×C42C2×D411yescharacteristic
215C2×C42C2×D411yescharacteristic
216C2×C42C2×D411yescharacteristic
217C2×C42#4118yes—
218C4⋊C8—44no—
219C22⋊C8—28no—
220C22⋊C8—44no—
221C22⋊C8—44no—
222C4×C8—44no—
223#95—28no—
224#95—24no—
225#95—24no—
226#95—24no—
227#95—24no—
228#95—28no—
229#95—44no—
230#95—44no—
231C24—22no—
232C24—22no—
233C24—22no—
234C24—22no—
235C24—41no—
236C24—41no—
237C24C2×C22⋊C411yescharacteristic
238C24C2×C22⋊C411yescharacteristic
239C24C2×D812yes—
240C24C2×SD1612yes—
241C24C22×D411yescharacteristic
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—216no—
260C22×C4—216no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—28no—
278C22×C4—24no—
279C22×C4—24no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4—41no—
283C22×C4—41no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4—41no—
287C22×C4—41no—
288C22×C4—41no—
289C22×C4—41no—
290C22×C4—416no—
291C22×C4—416no—
292C22×C4—44no—
293C22×C4—44no—
294C22×C4—44no—
295C22×C4—44no—
296C22×C4—44no—
297C22×C4—44no—
298C22×C4—44no—
299C22×C4—44no—
300C22×C4—44no—
301C22×C4—44no—
302C22×C4—44no—
303C22×C4—44no—
304C22×C4#10214yes—
305C22×C4#10214yes—
306C22×C4#10214yes—
307C22×C4#10214yes—
308C22×C4#10414yes—
309C22×C4C2×C22⋊C411yescharacteristic
310C22×C4C2×C22⋊C411yescharacteristic
311C22×C4C2×C22⋊C411yescharacteristic
312C22×C4C2×C22⋊C411yescharacteristic
313C22×C4C2×C22⋊C411yescharacteristic
314C22×C4C2×C22⋊C411yescharacteristic
315C22×C4C2×C22⋊C411yescharacteristic
316C22×C4C2×C22⋊C411yescharacteristic
317C22×C4C2×C22⋊C411yescharacteristic
318C22×C4C2×C22⋊C411yescharacteristic
319C22×C4#12314yes—
320C22×C4#12314yes—
321C22×C4C2×D812yes—
322C22×C4C2×D812yes—
323C22×C4C2×SD1612yes—
324C22×C4C2×SD1612yes—
325C22×C4#13714yes—
326C22×C4C23×C411yescharacteristic
327C22×C4C22×D411yescharacteristic
328C22×C4C2×#4114yes—
329C2×C8—44no—
330C2×C8—44no—
331C2×C8—44no—
332C4⋊C4—44no—
333C4⋊C4—44no—
334C4⋊C4—44no—
335C4⋊C4—44no—
336C4⋊C4—416no—
337C4⋊C4—44no—
338C4⋊C4—44no—
339C4⋊C4—44no—
340C4⋊C4—44no—
341C4⋊C4—44no—
342C4⋊C4—44no—
343C4⋊C4—44no—
344C4⋊C4—44no—
345C22⋊C4—416no—
346C22⋊C4—44no—
347C22⋊C4—44no—
348C22⋊C4—48no—
349C42—22no—
350C42—22no—
351C42—22no—
352C42—22no—
353C42—22no—
354C42—22no—
355C42—22no—
356C42—22no—
357C42—22no—
358C42—22no—
359C42—22no—
360C42—22no—
361C42—48no—
362C42—48no—
363C42—88no—
364C42#11714yes—
365C42#11714yes—
366C42#12314yes—
367C42#12314yes—
368C23—21no—
369C23—21no—
370C23—21no—
371C23—21no—
372C23—21no—
373C23—21no—
374C23—21no—
375C23—21no—
376C23—24no—
377C23—24no—
378C23—21no—
379C23—21no—
380C23—21no—
381C23—21no—
382C23—21no—
383C23—24no—
384C23—24no—
385C23—21no—
386C23—42no—
387C23—42no—
388C23—44no—
389C23—44no—
390C23—44no—
391C23—44no—
392C23#41012yes—
393C23C2×#10212yes—
394C23C2×#10212yes—
395C23C2×#10412yes—
396C23C22×C22⋊C411yescharacteristic
397C23C2×#12312yes—
398C23C22×D811yescharacteristic
399C23C22×SD1611yescharacteristic
400C23C2×#13712yes—
401C2×C4—28no—
402C2×C4—24no—
403C2×C4—24no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—28no—
419C2×C4—28no—
420C2×C4—28no—
421C2×C4—28no—
422C2×C4—28no—
423C2×C4—28no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—21no—
428C2×C4—21no—
429C2×C4—21no—
430C2×C4—44no—
431C2×C4—44no—
432C2×C4—44no—
433C2×C4—44no—
434C2×C4—44no—
435C2×C4—44no—
436C2×C4—44no—
437C2×C4—44no—
438C2×C4—44no—
439C2×C4—48no—
440C2×C4—48no—
441C2×C4—48no—
442C2×C4—44no—
443C2×C4—44no—
444C2×C4—44no—
445C2×C4—44no—
446C2×C4—44no—
447C2×C4—44no—
448C2×C4—84no—
449C2×C4—84no—
450C2×C4—84no—
451C2×C4—88no—
452C2×C4—84no—
453C2×C4—88no—
454C2×C4#38112yes—
455C2×C4#38112yes—
456C2×C4#38112yes—
457C2×C4#38112yes—
458C2×C4#41012yes—
459C2×C4C2×#10212yes—
460C2×C4C2×#10212yes—
461C2×C4C2×#10212yes—
462C2×C4C2×#10212yes—
463C2×C4#41912yes—
464C2×C4#41912yes—
465C2×C4C2×#10412yes—
466C2×C4C2×#10412yes—
467C2×C4#42112yes—
468C2×C4#48018yes—
469C2×C4#48318yes—
470C2×C4C2×#11712yes—
471C2×C4C2×#12312yes—
472C8—84no—
473C22—21no—
474C22—21no—
475C22—22no—
476C22—22no—
477C22—22no—
478C22—22no—
479C22—21no—
480C22—22no—
481C22—21no—
482C22—22no—
483C22—22no—
484C22—22no—
485C22—22no—
486C22—22no—
487C22—22no—
488C22—22no—
489C22—22no—
490C22—48no—
491C22—48no—
492C22—41no—
493C22—41no—
494C22—41no—
495C22—41no—
496C22#150214yes—
497C22C2×#38111yescharacteristic
498C22C2×#41011yescharacteristic
499C22C2×#41011yescharacteristic
500C22C22×#10211yescharacteristic
501C22C2×#41911yescharacteristic
502C22C22×#10411yescharacteristic
503C22C2×#42111yescharacteristic
504C22C2×#48012yes—
505C22C2×#48312yes—
506C4—22no—
507C4—22no—
508C4—22no—
509C4—22no—
510C4—22no—
511C4—22no—
512C4—22no—
513C4—41no—
514C4—41no—
515C4—88no—
516C4—84no—
517C4—84no—
518C4#182314yes—
519C4#182314yes—
520C4#182614yes—
521C2—22no—
522C2—22no—
523C2—24no—
524C2—42no—
525C2C2×#150211yescharacteristic
526C2#1754511yescharacteristic
527C2#1756211yescharacteristic
528C2#1757411yescharacteristic
529C2#2068811yescharacteristic
530C2C2×#182311yescharacteristic
531C2C2×#182611yescharacteristic
subgroupG/H
centerC22×C4C2×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC23C22×C22⋊C4
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C4 ◁ C23×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
4. #8716C2semidirect
6. #8014C2semidirect
14. #2285C4semidirect
15. #2285C4semidirect
18. C2×#384C4semidirect
19. C2×#384C4semidirect
25. C2×C4×C4⋊C4C4semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#22852
#207142
C4×#3802
#87162
#87122
#80142